For the horseback riding excursion where you ride in the ocean you should wear long pants. This is to prevent against chaffing while in the English saddles. We've worn swim suits under our clothing before (though some outfits actually have a place where you can change and can bring your swim gear in a bag). Another option that works well is wearing zip-offs. Before riding in the ocean you can simply remove the legs and then replace them afterwards. They'll give you ample time to prepare while they're getting the horses ready.
I thought I would share a funny story from a past CN trip where two couples we met went on the same horseback riding excursion. The 1st couple loved it, especially the ride through the ocean... and they highly recommended it. The 2nd couple hated it. They wore shorts and sandals and came back all scratched/chaffed up from the ride. I asked them if they enjoyed the ocean part like the other couple? They said no, explaining that the 1st couple rode up front but they rode near the back... and therefore they waded through tons of floating horse crap the whole time. They went into disquisting details about it and how they couldn't wait to get back so they could shower.
Anyway, I thought it was funny how two couples had such a different perspective on their horse back riding adventure. :-)
If you are going into the water, understand you will be up to your knees in the ocean (up to the horse's shoulder). We wore sneakers rather than swim shoes, and regretted it, because there is no way to keep the seawater out of the shoes. I recommend swim shoes or very old sneakers.
Pants are a good idea. There will likely be a place to change out of them (but don't expect a locker room - it will probably be a 3-sided stall). If you forget your pants (boy, if I had a nickle for every time.....) you can use a towel to shield your legs from the saddle.